1. Classification of Functions
We may classify functions by their formula as follows:
Polynomials
Linear Functions, Quadratic Functions. Cubic Functions.
Piecewise Defined Functions
Absolute Value Functions, Step Functions
Rational Functions
Algebraic Functions
Trigonometric and Inverse trigonometric Functions
Exponential Functions
Logarithmic Functions

2. Function’s Properties
We may classify functions by some of their properties as follows:
Injective (One to One) Functions
Surjective (Onto) Functions
Odd or Even Functions
Periodic Functions
Increasing and Decreasing Functions
Continuous Functions
Differentiable Functions

3. Power Functions

4. Combinations of Functions

5. Composition of Functions

6. Figure:

7. Inverse Functions

10. Exponential Functions

11. Logarithmic Functions

12. The logarithm with base e is called the natural logarithm and has a special notation:

13. Correspondence between degree and radian
The Trigonometric Functions
Correspondence between degree and radian

14. Trigonometric Identities
Some values of and
Trigonometric Identities

16. Graphs of the Trigonometric Functions

18. Inverse Trigonometric Functions
When we try to find the inverse trigonometric functions, we have a slight difficulty. Because the trigonometric functions are not one-to-one, they don’t have inverse functions. The difficulty is overcome by restricting the domains of these functions so that hey become one-to-one.

21. The Limit of a Function

27. Calculating Limits Using the Limit Laws

32. Infinite Limits; Vertical Asymptotes

34. Limits at Infinity; Horizontal Asymptotes

36. Tangents
The word tangent is derived from the Latin word tangens, which means “touching.”
Thus, a tangent to a curve is a line that touches the curve. In other words, a tangent line should have the same direction as the curve at the point of contact. How can his idea be made precise?
For a circle we could simply follow Euclid and say that a tangent is a line that intersects the circle once and only once. For more complicated curves this definition is inadequate.

40. Instantaneous Velocity; Average Velocity
If you watch the speedometer of a car as you travel in city traffic, you see that the needle doesn’t stay still for very long; that is, the velocity of the car is not constant. We assume from watching the speedometer that the car has a definite velocity at each moment, but how is the “instantaneous” velocity defined?
In general, suppose an object moves along a straight line according to an equation of motion , where is the displacement (directed distance) of the object from the origin at time . The function that describes the motion is called the position function of the object. In the time interval from to
the change in position is The average velocity over this time interval is

41. Now suppose we compute the average velocities over shorter and shorter time intervals In other words, we let approach . We define the velocity or instantaneous velocity at time to be the limit of these average velocities:
This means that the velocity at time is equal to the slope of the tangent line at .

44. The Derivative of a Function1

45. Differentiable Functions

46. The Derivative as a Function

48. What Does the First Derivative Function Say about the Original Function?

54. What Does the Second Derivative Function Say about the Original Function?

61. Indeterminate Forms and L’Hospital’s Rule

63. Antiderivatives